Exactly! This is what no one in the US seems to understand. My encounters with private clinics and hospitals in the UK (all 10+ years ago, at this point) were unbelievably luxurious, at prices that (totally, completely free-market driven, mind you) were affordable on middle-class incomes. Or, yeah: there's private medical insurance, also free-marketed to "shockingly reasonable", by US measures. Americans on good salaries have been bamboozled into believing that a single-payer system will trap them into some kind of hell-hole hospital° with no recourse, when in fact the exact opposite is true.
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°And, of course, the "hell-hole hospital" examples are cherry-picked. Bad on their own, of course, but not representative of a system as a whole, nor recognize that equally awful anecdotes are abundant in the USA.
I went to see my GP, paid for by public health, they referred me to a specialist.
I chose to pay €100 to see a private doctor who was available sooner (the next day) and had better ratings.
They referred me for an MRI which was done at another private provider, paid for by public health.
I went back to the private doctor and paid for a non-surgical treatment, which wasn't available on public health.
If that doesn't work, later I can opt for surgery, paid for by public health.
And even more importantly: There is one system that tracks all diagnoses, treatment, medication etc used by both public and private healthcare providers, so medical history is available instantly to everyone.